Lecture25 Ground Response Analysis Part1

8/12/2019 Lecture25 Ground Response Analysis Part1

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Ground Response Analysis

Part - I

Lecture-25

1



Source-Path-Site

Source

Path

SiteFault

2



Source-Path-Site

A complete ground response analysis should include:

•Rupture mechanism at source of an earthquake (source)

• Propagation of stress waves through the crust to the top of

bedrock beneath the site of interest (path)

• How ground surface motion is influenced by the soils that

lie above the bedrock (site)

3



Source-Path-Site

In reality,

•Mechanism of fault rupture is very complicated and

difficult to predict in advance

• Crustal velocity and damping characteristics are generally poorly known

• Nature of energy transmission between the source and site

is uncertain

4



Source-Path-Site

In Practice,

•Seismic hazard analyses (probabilistic or deterministic)

are used to predict bedrock motions at the location of the

site.

• Seismic hazard analyses rely on empirical attenuation

relationships to predict bedrock motion parameters.

• Ground response problem becomes one of determining

response of soil deposit to the motion of the underlying bedrock.

5



Objectives of ground response analysis

•Predict ground surface motions

Time histories

Response spectra

Scalar parameters

•Evaluate dynamic stresses and strains

Liquefaction hazards

Foundation loading•Evaluate ground failure potential

Instability of earth structures

Response of retaining structures 6



Rock Outcropping motion

Rock outcropping motion - the motion that would occur

where rock outcrops at a free surface

Rock

Soil

Rock outcroppingmotion

7



Bedrock motion

Bedrock motion - the motion that occurs at bedrock overlain by

a soil deposit. Differs from rock outcrop motion due to lack of

free surface effect.

Rock

Soil

Rock outcropping

motion

Bedrock motion 8



Free surface motion

Free surface motion - the motion that occurs at the surface of a

soil deposit.

Rock

Soil

Rock outcropping

motion

Bedrock motion

Free surface motion

9



Common problem 1

Rock outcrop motion is known - usually obtained from

attenuation relationship (based on database of rock outcrop

motions). Free surface motion is to be determined.

Rock

Soil

KnownUnknown

10



Common problem 2

Free surface motion is known - usually obtained from

attenuation relationship (based on database of soil outcrop

motions) Free surface motion is to be determined for site with

different soil conditions

Rock

Soil1

Known

Rock

Soil2

unknown

11



Ground response analysis

Two Basic Approaches:

12



Linear Ground response analysis

•A known time history of bedrock (input) motion is represented

as a Fourier series, usually using the FFT.

•Each term in the Fourier series of the bedrock (input) motion is

then multiplied by the transfer function to produce the Fourier

series of the ground surface (output) motion.

• The ground surface (output) motion can then be expressed in

the time domain using the inverse FFT.

13



Transfer function

•The transfer function determines how each frequency in the

bedrock (input) motion is amplified, or deamplified by the soil

deposit.

•A transfer function may be viewed as a filter that acts upon

some input signal to produce an output signal.

Transfer Function

(Filter)

Input Output

14



Transfer function Evaluation: cases

•Uniform undamped soil on rigid rock

•Uniform damped soil on rigid rock

•Uniform undamped soil on elastic rock

•Uniform damped soil on elastic rock

•Layered damped soil on elastic rock

Transfer Function

(Filter)

15



Uniform undamped soil on rigid rock

Assume harmonic base motion. Then, response should also be harmonic

u(z,t) = A ei(ωt + kz) + B ei(ωt - kz)

Wave traveling

in -z direction

(upward)

Wave traveling

in +z direction

(downward)

H

u

z A ei( t+kz)

B ei( t-kz)

2

2

2

2

z

uG

t

u

Wave Equation :

16




Displacement


Stress

t(z,t) = Gg(z,t) = GikA ei(ωt + kz) - GikB ei(ωt - kz)

At z = 0 (ground surface)

t(z,t) = 0 = Gik(A - B) eiωt

A = B

H

u

z A ei( t+kz)

B ei( t-kz)

17




Displacement


Substituting A = B

u(z,t) = 2A [(eikz

+ e-ikz

)/2] eiωt

= 2A cos(kz) eiωt

Defining a transfer function as the ratio of the displacement

at the ground surface to the bedrock displacement

kH ekH A

Ae

t H u

t uF

t i

t i

cos)cos(),(

),()(

1

2

20

18




The modulus of the transfer function is the amplification function(|F()|)

kH ekH A

Ae

t H u

t uF

t i

t i

cos)cos(),(

),()(

1

2

20

sV H kH

F

coscos

)( 11

19




As kH = H/Vs goes to zero, denominator goes to zero and the transfer function

goes to infinity, causing resonance. From this condition, the natural frequency

(n) and the fundamental period (TS)can be estimated as:

Natural Frequencies:

H nV S n 2

Fundamental Period

S s V H T 42 0

20



Uniform damped soil on rigid rock

Damping is handled by Complex shear modulus G*

G* = ρ (v* s )2 = ρ (ω/k*)2

k* = [ρω2 /G*] 1/2 Complex wave number

)1(/** ivGv s s )1(*

*

ik v

k

s

H

u z A ei( t+kz)

B ei( t-kz) Damped

21




Transfer function for this case is evaluated as it is done

for the case of undamped soil. The equation for

transfer function now becomes:

H

u z A ei( t+kz)

B ei( t-kz) Damped

H k e H k A

Ae

t H u

t uF

t i

t i

**cos)cos(),(

),(),(

1

2

20

22




*

*

coscos),(

),(),(

sV

H H k t H u

t uF

110

2222

11

ss V H V H kH kH

F

//cos)()(cos

),(

23




Observations:

Natural frequencies still exist

Amplification is strongly dependent

on the frequencies

Low natural frequencies are amplified

High natural frequencies are weaklyamplified

Very high frequencies are de-

amplified

24



25

Kramer (1996) Geotechnical Earthquake Engineering, Prentice Hall.

Villaverde, R. (2009) Fundamental Concepts of Earthquake Engineering , CRC

Press.

Towhata, T. (2008) Geotechnical Earthquake Engineering, Springer.

References

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Lecture25 Ground Response Analysis Part1

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